Question: All of the 5th grade teachers and students from Covington went on a field trip to an archaeology museum. Tickets were $$8.00$ each for teachers and $$3.50$ each for students, and the group paid $$51.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$16.00$ each for teachers and $$7.50$ each for students, and the group paid $$107.00$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8x+3.5y = 51}$ ${16x+7.5y = 107}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-16x-7y = -102}$ ${16x+7.5y = 107}$ Add the top and bottom equations together. $ 0.5y = 5 $ $ y = \dfrac{5}{0.5}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {8x+3.5y = 51}$ to find $x$ ${8x + 3.5}{(10)}{= 51}$ $8x+35 = 51$ $8x = 16$ $x = \dfrac{16}{8}$ ${x = 2}$ You can also plug ${y = 10}$ into $ {16x+7.5y = 107}$ and get the same answer for $x$ ${16x + 7.5}{(10)}{= 107}$ ${x = 2}$ There were $2$ teachers and $10$ students on the field trips.